The YouTube logo and context menu can be configured by setting additional parameters to the video source. However, this is currently not supported by the RadMediaPlayer control.
I have logged a feature request so that the functionality can be included in a future release of the controls. In order to increase the priority of the feature you can vote for it in the link below.
Do you want to have your say when we set our development plans?
Do you want to know when a feature you care about is added or when a bug fixed?
Telerik Feedback Portal
and vote to affect the priority of the items